Pairs of mutually annihilating operators

Abstract

Pairs ( A , B ) of mutually annihilating operators AB = BA = 0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1–58] by method of linear relations. The classification of ( A , B ) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636–654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of ( A , B ) over any field in an explicit form and our proof is constructive: the matrices of ( A , B ) are sequentially reduced to their canonical form by similarity transformations ( A , B ) ↦ ( S - 1 AS , S - 1 BS ) .